This paper focuses on the analysis of conforming virtual element methods for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward problem, when the source term belongs to $H^{-1}(\Omega)$, the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming virtual element method. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming virtual element method and allows data in $H^{-1}(\Omega)$. This paper has {\it three} main contributions. The {\it first} contribution is the design of a conforming companion operator $J$ from the {\it conforming virtual element space} to the Sobolev space $V:=H^1_0(\Omega)$, a modified virtual element scheme, and the \textit{a priori} error estimate for the Poisson problem in the best-approximation form without data oscillations. The {\it second} contribution is the extension of the \textit{a priori} analysis to general second-order elliptic problems with source term in $V^*$. The {\it third} contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to $V^*$. The Tikhonov's regularization technique regularizes the ill-posed inverse problem, and the conforming virtual element method approximates the regularized problem given a finite measurement data. The inverse problem is also discretised using the conforming virtual element method and error estimates are established. Numerical tests on different polygonal meshes for general second-order problems, and for a Poisson inverse source problem with finite measurement data verify the theoretical results.

翻译：本文针对具有粗糙源项的一般二阶线性椭圆问题，分析了相容虚拟元方法，并将其应用于含粗糙观测数据的泊松逆源问题。对于正问题，当源项属于$H^{-1}(\Omega)$时，标准相容虚拟元方法中通过多项式投影定义的离散近似右端项失去意义。本文提出的修正离散格式在相容虚拟元方法框架下引入新型伴算子，使得数据可属于$H^{-1}(\Omega)$。论文有**三个**主要贡献：**第一**贡献是设计了一个从**相容虚拟元空间**到Sobolev空间$V:=H^1_0(\Omega)$的相容伴算子$J$，构建了修正虚拟元格式，并给出了泊松问题在无数据振荡最优逼近形式下的**先验**误差估计；**第二**贡献是将该先验分析推广至源项属于$V^*$的一般二阶椭圆问题；**第三**贡献是将伴算子应用于观测数据属于$V^*$的泊松逆源问题。采用吉洪诺夫正则化技术对不适定逆问题进行正则化处理，相容虚拟元方法在给定有限观测数据条件下逼近正则化问题。同时使用相容虚拟元方法离散逆问题，并建立了误差估计。针对一般二阶问题及有限观测数据泊松逆源问题的多边形网格数值试验验证了理论结果。